A NOTE ON THE COHOMOLOGY OF FINITE DIMENSIONAL COCOMMUTATIVE HOPF ALGEBRAS

JOHN H. PALMIERI

June 2, 1994

Abstract. In the context of finite dimensional cocommutative Hopf algebras, we prove versions of various group cohomology results: the Quillen-Venkov theorem on detecting nilpotence in group cohomology, Chouinard’s theorem on determining whether a kG-module is projective by restricting to elementary abelian p-subgroups of G, and Quillen’s theorem which identifies the cohomology of G, “modulo nilpotent elements.” This last result is only proved for graded connected Hopf algebras.

1. Introduction

When one studies the mod p cohomology of finite groups, one sees that elementary abelian p-subgroups play an important role. In this paper we try to generalize this to finite dimensional cocommutative Hopf algebras; Wilkerson laid the foundations for this work in his paper [14]. Since the setting is more general, the analogues of elementary abelian subgroups are not as pleasant.

Unless otherwise indicated, every Hopf algebra in this paper will be finite dimensional graded cocommutative and defined over a field k of characteristic p > 0. We require cocommutativity for two reasons: so that the cohomology is a commutative k-algebra, and so that the Hopf algebra is free as a left module over every sub-Hopf algebra (see Radford [12]).

Definition 1.1. A Hopf algebra C over k is elementary if it is bicommutative and has x^{p} = 0 for all x ∈ IC, the augmentation ideal of C. We will use C(x) to denote a monogenic elementary Hopf algebra, generated by x; i.e., C(x) is isomorphic as an algebra to k[x]/(x^{n}), where n is p or 2. For a Hopf algebra E, we call a nonzero element v ∈ Ext_{E}^{1}(k, k) an extension element if the corresponding extension of augmented algebras,

E′ → E → C(x),

is an extension of Hopf algebras. Then Ext_{C(x)}^{1}(k, k) is one-dimensional, and v is the image of a nonzero class; we denote by β′(v) ∈ Ext_{E}^{∗}(k, k) the image of the polynomial generator of Ext_{C(x)}^{∗}(k, k). A Hopf algebra E is quasi-elementary if no product of β′(v)’s is nilpotent. (See Lemma A.1 for an alternate description of extension elements.)

1991 Mathematics Subject Classification. Primary 16W30,18G15, 57T05; Secondary 20J06.

In the p-group setting, “elementary” and “quasi-elementary” are the same notion by a theorem of Serre [13]—they both correspond to the group algebras of elementary abelian p-groups—but they differ for more general Hopf algebras. Wilkerson [14] has shown that for every p, there is a (graded connected) non-elementary quasi-elementary Hopf algebra E over a field of characteristic p. For cohomological purposes, quasi-elementary Hopf algebras carry more information; our main theorems illustrate this. In other words, quasi-elementary Hopf algebras seem to be the correct analogue of elementary abelian p-groups. As such, it would be nice to have alternate characterizations of them; see the appendix for a few examples.

We move on to the statements of the main results. These are analogous to group cohomology theorems of Carlson [3], Chouinard [4], and Quillen [11]. The first is a generalization of a theorem of Wilkerson [14]; see also [9] (and we’ve patterned the statement on the nilpotence theorem of [5]). For all of these, B is a finite dimensional graded cocommutative Hopf algebra over k, char(k) = p.

For a class z ∈ Ext_{B}^{∗}(L, M), we say that z is tensor-nilpotent if the element z^{⊗n} ∈ Ext_{B}^{∗}(L^{⊗n}, M^{⊗n}) is zero for n sufficiently large.

Theorem 1.2. (a) Let Γ be an associative B-algebra with unit η : k → Γ; fix z ∈ Ext_{B}^{∗}(k, Γ). Then z is nilpotent if and only if res_{B,E}(z) is nilpotent for every quasi-elementary sub-Hopf algebra E of B.

(b) Let M be a B-module; fix z ∈ Ext_{B}^{∗}(M, M). Then z is nilpotent if and only if res_{B,E}(z) is nilpotent for every quasi-elementary sub-Hopf algebra E of B.

(c) Let L and M be B-modules; fix z ∈ Ext_{B}^{∗}(L, M). Then z is tensor-nilpotent if and only if res_{B,E}(z) is tensor-nilpotent for every quasi-elementary sub-Hopf algebra E of B.

Theorem 1.3. Let M be a B-module. Then M is projective if and only if M restricted to E is projective for every quasi-elementary sub-Hopf algebra E of B.

Theorem 1.4. Suppose that B is connected. Let C_{B} be the category with objects the quasi-elementary sub-Hopf algebras of B, and morphisms given by inclusions. Then the natural map

q_{B} : Ext_{B}^{∗}(k, k) → \underleftarrow{lim}_{E∈C_{B}} Ext_{E}^{∗}(k, k)

is an isomorphism mod nilpotents: the range is finitely generated as a module over Ext_{B}^{∗}(k, k), every element in the kernel of q_{B} is nilpotent, and for every z in the range, there is an integer n so that z^{p^{n}} is in the image of q_{B}.

Regarding Theorem 1.4, in the corresponding statement for a group G, the objects in the category C_{G} are elementary abelian p-subgroups E of G, and the morphisms are given by inclusions and conjugations (and composites of these). In the graded connected Hopf algebra case, we don’t need conjugations. One reason for this is that in our setting, conjugations always turn out to be inclusions. Also, in the group case, given E elementary abelian, the conjugations are needed to reflect the action on H^{∗}(E) by the normalizer of E—the image of the restriction map from H^{∗}(G) is contained in the invariants. By [6, Theorem 4.12], though, in the Hopf algebra case all of H^{∗}(E) is in the image of the restriction map, up to nilpotence.

It would be nice to remove the connectedness condition in Theorem 1.4; currently, though, it is not even known if Ext_{B}^{∗}(k, k) is a finitely generated algebra, so it may be unreasonable to ask for more precise information. Our proof will certainly not generalize, because we deduce Theorem 1.4 from the work of Hopkins and Smith, and they make heavy use of connectedness. In any case, we have the following conjecture.

Conjecture 1.5. Let B be a finite dimensional cocommutative Hopf algebra; let C_{B} be as above. The natural map

q_{B} : Ext_{B}^{∗}(k, k) → \underleftarrow{lim}_{E∈C_{B}} Ext_{E}^{∗}(k, k)

is an isomorphism mod nilpotents.

The structure of the paper is as follows: in Section 2 we explain our notation and review a few facts about Hopf algebras; in Section 3 we prove Theorems 1.2 and 1.3; we prove Theorem 1.4 in Section 4; and we give some examples of quasi-elementary Hopf algebras in the appendix.

Acknowledgements: I would like to thank Haynes Miller and Clarence Wilkerson for providing the inspiration for the above definition of quasi-elementary; indeed, Theorem 1.2 is essentially their result, at least for M = k (private communication). Example A.6 is also due to them. I had several quite profitable discussions about Hopf algebras with Mark Feshbach, as well.

2. Notation and conventions

Throughout the paper all Hopf algebras are graded, cocommutative and defined over a field k of characteristic p > 0. Commutative and cocommutative are in the usual graded sense. We use two particular Hopf algebras several times:

E[x] = k[x]/(x^{2}),
D[x] = k[x]/(x^{p}).

(Note that if p is odd, then E[x] has a Hopf algebra structure only if x is in an odd grading.) The coalgebra structure will vary, depending on the context; two common choices are that Δ(x) = 1 ⊗ x + x ⊗ 1 (this is the only choice, if x is in a positive grading), and Δ(x) = 1⊗x+x⊗x+x⊗1. We will also use C(x) to denote either of these (so C(x) is an arbitrary monogenic elementary Hopf algebra).

If B is a Hopf algebra, we use H^{∗}(B) to denote Ext_{B}^{∗}(k, k), as usual. The augmentation ideal of B is written IB. We write A ≤ B to mean that A is a sub-Hopf algebra of B; similarly, A ◁ B means that A is a normal sub-Hopf algebra of B (so IA · B = B · IA). If A ≤ B, we write res_{B,A} for the restriction map

res_{B,A} : Ext_{B}^{∗}(−, −) → Ext_{A}^{∗}(−, −).

If A◁B, then B//A is the quotient Hopf algebra B ⊗_{k} A = B/(B ·IA). If χ denotes the anti-automorphism of B, then we can define conjugation by b ∈ B to be the map

B ⟶^{c_{b}} B,
a ↦ ∑_{i} b′_{i}aχ(b′′_{i})

where b has coproduct ∑_{i} b′_{i} ⊗ b′′_{i}. Then a sub-Hopf algebra A of B is normal if and only if c_{b}(a) ∈ A for all a ∈ A, b ∈ B (see [10]). There are a number of other similarities between normal sub-Hopf algebras and normal subgroups; we will need the following: if A, A′ ◁B, then AA′ ≤ B; furthermore, we have A∩A′ ◁A′, A◁AA′, and there is an isomorphism

A′//(A ∩ A′) ≅ (AA′)//A.

(The proof is a straightforward verification.)

Note that for any cocommutative Hopf algebra B, the cohomology of B has an action of the Steenrod algebra (see May [8, Section 11]); we use May’s notation (see also [14]) for indexing of Steenrod operations on Hopf algebra cohomology.

3. Proofs of Theorems 1.2 and 1.3

We prove Theorems 1.2 and 1.3 by imitating proofs for the group theory analogues. We start with the following definition; this replaces the product of the Bocksteins of the one-dimensional classes in group cohomology.

Definition 3.1. Suppose B is not quasi-elementary, so that there is some product of β′(v)’s, for extension elements v ∈ Ext_{B}^{1}(k, k), which is nilpotent. Define the fundamental class of B, u_{B}, to be any such product. (This is far from unique, of course.)

The following is our key lemma, analogous to [14, 5.2] and [2, Vol. II, 5.2.1].

Lemma 3.2. Suppose v is an extension element, corresponding to the extension

A → B → C(x).

Let Γ be an associative B-algebra with unit η : k → Γ, and fix z ∈ Ext_{B}^{∗}(k, Γ).

(a) Suppose C(x) = E[x]. If res_{B,A}(z) = 0, then z = η_{∗}(β′(v))z′ for some z′.

Hence z^{2} = η_{∗}(β′(v))z′′ for some z′′.

(b) Suppose C(x) = D[x], with p odd. If res_{B,A}(z) = 0, then z^{2} = η_{∗}(β′(v))z′′

for some z′′.

Proof. This is proved just as in [2]; Benson attributes the proof to Kroll [7]. We prove part (a) and leave part (b) to the reader. Note that in the setting of part (a), we have β′(v) = v. (For part (b), β′(v) = βP̃^{0}(v).)

There is a short exact sequence of E[x]-modules, and hence of B-modules:

0 → k → E[x] ⟶^{q} k → 0. (1)

This module extension corresponds to v ∈ Ext_{B}^{1}(k, k). Note that the composite

Ext_{B}^{∗}(k, Γ) ⟶^{q^{∗}} Ext_{B}^{∗}(E[x], Γ) ≅ Ext_{A}^{∗}(k, Γ)

is the restriction map—see [2, Vol. I, Section 2.8]. Consider the long exact sequence in Ext_{B}^{∗}(−, Γ) associated to the extension (1): since q^{∗}(z) = 0 then z is in the image of the boundary homomorphism; but the boundary homomorphism is multiplication by η_{∗}(v). Since this element is central in Ext_{B}^{∗}(k, Γ), the result follows.

Proof of Theorem 1.2. First we prove part (a); the other parts will follow from this.

This is proved by induction on the sub-Hopf algebras of B using Lemma 3.2, just as in [14]. If A ≤ B and dim_{k}(A) = 1, then A = k and the theorem is trivially true for z ∈ Ext_{A}^{∗}(k, Γ). This starts the induction.

Suppose the theorem holds for all proper sub-Hopf algebras A of B, and suppose z ∈ Ext_{B}^{∗}(k, Γ) is nilpotent upon restriction to every quasi-elementary sub-Hopf algebra of B. We may assume that B is not quasi-elementary, so that B has a fundamental class u_{B}. For each factor β′(v) in u_{B}, we have an extension

A → B → C(x).

By induction, we know that res_{B,A}(z) is nilpotent, and so we may replace z with a large enough power so that res_{B,A}(z) = 0. Hence z^{2} = η_{∗}(β′(v))z′′ for some z′′. Since this is true for every factor of u_{B}, we see that z^{r} = η_{∗}(u_{B})ζ for some integer r and some ζ ∈ Ext_{B}^{∗}(k, Γ). Since η_{∗}(u_{B}) is nilpotent and central in Ext_{B}^{∗}(k, Γ), then z is nilpotent.

Part (b) follows from (a) via the algebra isomorphism

Ext_{B}^{∗}(M, M) ≅ Ext_{B}^{∗}(k, Hom_{k}(M, M)).

Part (c) also follows easily: we have Ext_{B}^{∗}(L, M) ≅ Ext_{B}^{∗}(k, Hom_{k}(L, M)), so let Γ be the tensor-algebra of the B-module Hom_{k}(L, M):

Γ = ⨁_{n≥0}(Hom_{k}(L, M))^{⊗n}.

Then z ∈ Ext_{B}^{∗}(L, M) is tensor-nilpotent if and only if the corresponding element z̃ ∈ Ext_{B}^{∗}(k, Γ) is nilpotent.

Remark 3.3. The mod 2 Steenrod algebra A is a union of finite dimensional sub-Hopf algebras, and one can prove Theorem 1.2 for A (see [9]). (Note that there are no extension elements in Ext_{A}^{∗}(F_{2}, F_{2}); so in this context (graded, connected, possibly infinite dimensional Hopf algebras), we add to the definition of quasielementary that there exist extension elements in Ext_{E}^{1}(k, k).) It may be worthwhile to examine detection of nilpotence in Ext_{B}, for any cocommutative Hopf algebra B which is the union of finite dimensional sub-Hopf algebras.

We move on to a discussion of Theorem 1.3.

Lemma 3.4. Let B, v, and A be as in Lemma 3.2. Let M and N be B-modules. Define the function

μ : Ext_{B}^{∗}(M, N) → Ext_{B}^{∗}(M, N)

to be composition with β′(v) ⊗ 1_{M}. If the restriction of M to A is projective, then μ is an isomorphism in positive degrees.

Proof. This is proved as in [4]; namely, we observe that the spectral sequence

Ext_{C(x)}^{∗}(k, Ext_{A}^{∗}(M, N)) ⇒ Ext_{B}^{∗}(M, N)

associated to the extension

A → B → C(x)

collapses.

Proof of Theorem 1.3. First, if M is projective over B, then by a result of Radford [12], M is projective over every E ≤ B. The converse is proved as in [4]: we may assume that B is not quasi-elementary; let u_{B} be a fundamental class for B. Fix N an arbitrary B-module. By induction on the factors of u_{B} and Lemma 3.4, multiplication by u_{B} ⊗ 1_{M} is an isomorphism in Ext_{B}^{s}(M, N) for s > 0. But u_{B} is nilpotent; hence Ext_{B}^{s}(M, N) = 0 for s > 0. Since this holds for all B-modules N, then Hom_{B}(M, −) is exact, and so M is projective over B.

Remark 3.5. For the sub-Hopf algebras of the mod 2 Steenrod algebra A, the notions of elementary and quasi-elementary are the same—see [14]. So Theorem 1.3 provides an almost immediate proof of a theorem of Adams and Margolis [1], that a bounded below A-module M is free if and only if H(M, P_{t}^{s}) = 0 for all P_{t}^{s} ∈ A with s < t. Indeed, their theorem is basically a special case of Theorem 1.3, combined with [14, Theorem 6.4]. Here is a related conjecture; this would provide a replacement for the Adams-Margolis theorem when working with non-bounded below modules.

Conjecture 3.6. Theorem 1.3 holds for arbitrary sub-Hopf algebras of the mod 2 Steenrod algebra. In particular, it holds for the full Steenrod algebra.

4. Proof of Theorem 1.4

Let B be a graded connected cocommutative Hopf algebra over a field k of characteristic p. Our goal here is to prove Theorem 1.4, the analogue of the Quillen stratification theorem. It follows from Wilkerson’s work on finite generation in [14], Theorem 1.2 for M = k, and the next result.

Theorem 4.1. Let C be a poset of sub-Hopf algebras of B which is closed under inclusions—if A ∈ C and A′ ≤ A, then A′ ∈ C. Then the natural map

q : Ext_{B}^{∗}(k, k) → \underleftarrow{lim}_{A∈C} Ext_{A}^{∗}(k, k)

is surjective mod nilpotents: for every z in the range, there is an integer n so that z^{p^{n}} is in the image of q.

This in turn essentially follows from a theorem of Hopkins-Smith [6, Theorem 4.12]: if A is a sub-Hopf algebra of B and z ∈ H^{∗}(A), then for some n, z^{p^{n}} is in the image of res_{B,A}.

Proof. The goal here is to show that if we have a compatible family of elements z_{A} ∈ H^{∗}(A) for each A ∈ C, then there is an element y ∈ H^{∗}(B) and an integer n so that res_{B,A}(y) = (z_{A})^{p^{n}} for each A.

By induction on the sub-Hopf algebras of B, it suffices to consider the case where C is the collection of all proper sub-Hopf algebras of B: let B′ be a minimal sub-Hopf algebra of B not in C; then C contains all of the proper sub-Hopf algebras of B′, so there is an element z of H^{∗}(B′) compatible with powers of the other z_{A}’s, and hence we can add B′ to C (after replacing the z_{A}’s with large enough powers—note that each B′ has finitely many sub-Hopf algebras).

We have the following lemmas.

Lemma 4.2. If A is a maximal sub-Hopf algebra of B, then A is normal and B//A ≅ C[u]. Slightly more generally, if A ≤ B and A′ ≤ B are maximal with A ≠ A′, then

A′//(A ∩ A′) ≅ B//A ≅ C[u].

Proof. For the first statement, see [6, Lemma A.11]. To identify the quotient Hopf algebra in the second statement, note that AA′ = B—since A, A′ ◁ B, then AA′ is a sub-Hopf algebra of B, and it properly contains the maximal sub-Hopf algebra A. Now, use the isomorphism given in Section 2:

A′//(A ∩ A′) ≅ AA′//A.

Let A_{1}, A_{2}, …, A_{n} be the (distinct) maximal elements of C. Then for each i we have a Hopf algebra extension

A_{i} → B → C[u_{i}].

Now, H^{∗}(C[u_{i}]) is either k[v_{i}] or k[w_{i}] ⊗ E[v_{i}]; to simplify notation, we use β_{i} to denote the image of the polynomial generator in H^{∗}(B) (so β_{i} is β′(v_{i}), in our earlier notation).

Lemma 4.3. With the above notation, we have (β_{i}) ⊆ ker(res_{B,A_{i}}) ⊆ \sqrt{(β_{i})}. Slightly more generally, we have

(β_{i}) + (β_{j}) ⊆ ker(res_{B,A_{i}∩A_{j}}) ⊆ \sqrt{(β_{i}) + (β_{j})}.

Proof. The statement about ker(res_{B,A_{i}}) is the content of [14, Proposition 5.2], or equivalently Lemma 3.2. It immediately follows that (β_{i}) + (β_{j}) is contained in ker(res_{B,A_{i}∩A_{j}}). Suppose x ∈ ker(res_{B,A_{i}∩A_{j}}). Then

res_{B,A_{i}}(x) ∈ ker(res_{A_{i},A_{i}∩A_{j}}).

If we can show that ker(res_{A_{i},A_{i}∩A_{j}}) ⊆ \sqrt{(res_{B,A_{i}}(β_{j}))}, we are done. We invoke Lemma 4.2: since A_{i}//(A_{i}∩A_{j}) ≅ C[u_{j}], then ker(res_{A_{i},A_{i}∩A_{j}}) is contained in \sqrt{(β)}, where β is the homomorphic image of the polynomial generator in H^{∗}(A_{i}//A_{i} ∩A_{j}). This β is precisely res_{B,A_{i}}(β_{j}).

For each i, we have an element z_{i} ∈ H^{∗}(A_{i}); by the Hopkins-Smith theorem, we can replace each z_{i} by z_{i}^{p^{n}} so that there is an element x_{i} ∈ H^{∗}(B) which restricts to z_{i}. There are many choices for such an x_{i}—by Lemma 4.3, any element in the coset x_{i} + (β_{i}) will do. We want to find a single element x which restricts to each z_{i}. We may raise everything to a large power; this means that we want to show that for some r, the set

⋂_{i=1}^{n}(x_{i}^{p^{r}} + (β_{i}))

is nonempty. We do this by induction: clearly, x_{1} + (β_{1}) is nonempty. Assume we have an element y ∈ ⋂_{i=1}^{k−1}(x_{i} + (β_{i})). For some r, we want an element in

⋂_{i=1}^{k}(x_{i}^{p^{r}} + (β_{i})) = ( ⋂_{i=1}^{k−1}(y^{p^{r}} + (β_{i}))) ∩ (x_{k}^{p^{r}} + (β_{k})),

so it certainly suffices to find an element in

(y^{p^{r}} + (β_{1}β_{2} … β_{k−1})) ∩ (x_{k}^{p^{r}} + (β_{k})).

Since y and x_{k} restrict compatibly to A_{i} ∩ A_{k}, then y − x_{k} ∈ ker(res_{B,A_{i}∩A_{k}}) for each i. By Lemma 4.3, if we replace y and x_{k} by y^{p^{r}} and x_{k}^{p^{r}} for r large enough, then for each i we have elements a_{i} and b_{i} so that

y − x_{k} − b_{i}β_{k} = a_{i}β_{i}. (2)

Suppose p^{r} is the smallest power of p at least as big as k − 1; then we multiply the equations (2) together (with the first with multiplicity p^{r} − k + 2) to get

y^{p^{r}} − x_{k}^{p^{r}} + cβ_{k} = dβ_{1}β_{2} … β_{k−1}

for some elements c and d. This equality shows that the desired intersection is nonempty, and so completes the induction.

Appendix A. Some quasi-elementary Hopf algebras

The above definitions of extension elements and quasi-elementary Hopf algebras are a bit on the abstract side. In this appendix, we recall from [14, Proposition 5.1] an alternate characterization of extension elements of positive internal degree, and we give some examples of quasi-elementary Hopf algebras. Let B be a graded cocommutative Hopf algebra over a field k of characteristic p.

Lemma A.1 ([14]). Fix a nonzero v ∈ Ext_{B}^{1,n}(k, k), with n > 0.

(a) Suppose p = 2. Then v is an extension element if and only if v ∈ ker S̃q^{0}.

(b) Suppose p is odd. Then v is an extension element if and only if n is odd or v ∈ ker P̃^{0}.

(Wilkerson proves, for example, that if v ∈ ker S̃q^{0}, then v is an extension element; the converse is immediate.) The following seems like a reasonable compromise between the above case and the group algebra case (in which S̃q^{0} acts as the identity). We don’t have enough evidence to grace this with the label of “Conjecture.”

Guess A.2. Fix v ∈ Ext_{B}^{1,0}(k, k). For p = 2 (respectively, p odd), v is an extension

element if and only if S̃q^{0}(v) (resp., P̃^{0}(v)) is a scalar multiple of v.

We conclude with a number of examples of quasi-elementary Hopf algebras. These are, for the most part, reasonably nice Hopf algebras; however, there is no reason to expect all quasi-elementary Hopf algebras to be so nicely behaved. We would be (pleasantly) surprised if there were a nice element-wise characterization in general. Having a result like A.2 would be a step, maybe even a step in the right direction.

Example A.3. Suppose E is a p-group; then kE is quasi-elementary if and only if E is elementary abelian. This is the content of Serre’s theorem on products of Bocksteins [13]. We should also point out that if G is a p-group, then every v ∈ Ext_{kG}^{1}(k, k) is an extension element.

Example A.4. Suppose E is the universal enveloping algebra of a restricted Lie algebra. It is possible for E to be quasi-elementary, but not elementary. See [14, Example 6.5].

Example A.5. Suppose E is a finite dimensional sub-Hopf algebra of the mod 2 Steenrod algebra. Then E is quasi-elementary if and only if E is elementary—see [14, Theorem 6.4]. In terms of the dual of A, this means that for some m ≥ 1, E is a sub-Hopf algebra of

(F_{2}[ξ_{m}, ξ_{m+1}, ξ_{m+2}, …]/(ξ_{m}^{2^{m}}, ξ_{m+1}^{2^{m}}, ξ_{m+2}^{2^{m}}, …))^{∗} .

Example A.6. Let p be an odd prime, and suppose E is a finite dimensional sub-Hopf algebra of the mod p reduced powers. Then E is quasi-elementary if and only if, for some m ≥ 2, E is a sub-Hopf algebra of

(F_{p}[ξ_{m−1}, ξ_{m}, ξ_{m+1}, ξ_{m+2}, …]/(ξ_{m−1}^{p}, ξ_{m}^{p^{m}}, ξ_{m+1}^{p^{m}}, ξ_{m+2}^{p^{m}}, …))^{∗} .

See [14], [9].

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Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA E-mail address: palmieri@math.mit.edu